Optimal. Leaf size=241 \[ -\frac {2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^{3/2} f^{3/2}}+\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d f}+\frac {1}{3} x^3 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {i b n \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2}}-\frac {i b n \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2}}+\frac {2 b n \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{9 d^{3/2} f^{3/2}}-\frac {1}{9} b n x^3 \log \left (d f x^2+1\right )-\frac {8 b n x}{9 d f}+\frac {4}{27} b n x^3 \]
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Rubi [A] time = 0.18, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {2455, 302, 205, 2376, 4848, 2391, 203} \[ \frac {i b n \text {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2}}-\frac {i b n \text {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2}}-\frac {2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^{3/2} f^{3/2}}+\frac {1}{3} x^3 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d f}-\frac {2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {2 b n \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{9 d^{3/2} f^{3/2}}-\frac {1}{9} b n x^3 \log \left (d f x^2+1\right )-\frac {8 b n x}{9 d f}+\frac {4}{27} b n x^3 \]
Antiderivative was successfully verified.
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Rule 203
Rule 205
Rule 302
Rule 2376
Rule 2391
Rule 2455
Rule 4848
Rubi steps
\begin {align*} \int x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx &=\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d f}-\frac {2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^{3/2} f^{3/2}}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-(b n) \int \left (\frac {2}{3 d f}-\frac {2 x^2}{9}-\frac {2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2} x}+\frac {1}{3} x^2 \log \left (1+d f x^2\right )\right ) \, dx\\ &=-\frac {2 b n x}{3 d f}+\frac {2}{27} b n x^3+\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d f}-\frac {2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^{3/2} f^{3/2}}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\frac {1}{3} (b n) \int x^2 \log \left (1+d f x^2\right ) \, dx+\frac {(2 b n) \int \frac {\tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{x} \, dx}{3 d^{3/2} f^{3/2}}\\ &=-\frac {2 b n x}{3 d f}+\frac {2}{27} b n x^3+\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d f}-\frac {2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^{3/2} f^{3/2}}-\frac {1}{9} b n x^3 \log \left (1+d f x^2\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {(i b n) \int \frac {\log \left (1-i \sqrt {d} \sqrt {f} x\right )}{x} \, dx}{3 d^{3/2} f^{3/2}}-\frac {(i b n) \int \frac {\log \left (1+i \sqrt {d} \sqrt {f} x\right )}{x} \, dx}{3 d^{3/2} f^{3/2}}+\frac {1}{9} (2 b d f n) \int \frac {x^4}{1+d f x^2} \, dx\\ &=-\frac {2 b n x}{3 d f}+\frac {2}{27} b n x^3+\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d f}-\frac {2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^{3/2} f^{3/2}}-\frac {1}{9} b n x^3 \log \left (1+d f x^2\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {i b n \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2}}-\frac {i b n \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2}}+\frac {1}{9} (2 b d f n) \int \left (-\frac {1}{d^2 f^2}+\frac {x^2}{d f}+\frac {1}{d^2 f^2 \left (1+d f x^2\right )}\right ) \, dx\\ &=-\frac {8 b n x}{9 d f}+\frac {4}{27} b n x^3+\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d f}-\frac {2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^{3/2} f^{3/2}}-\frac {1}{9} b n x^3 \log \left (1+d f x^2\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {i b n \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2}}-\frac {i b n \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2}}+\frac {(2 b n) \int \frac {1}{1+d f x^2} \, dx}{9 d f}\\ &=-\frac {8 b n x}{9 d f}+\frac {4}{27} b n x^3+\frac {2 b n \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{9 d^{3/2} f^{3/2}}+\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d f}-\frac {2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^{3/2} f^{3/2}}-\frac {1}{9} b n x^3 \log \left (1+d f x^2\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {i b n \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2}}-\frac {i b n \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 364, normalized size = 1.51 \[ -\frac {2 a \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2}}+\frac {1}{3} a x^3 \log \left (d f x^2+1\right )+\frac {2 a x}{3 d f}-\frac {2 a x^3}{9}-\frac {2 b \left (3 \left (\log \left (c x^n\right )-n \log (x)\right )-n\right ) \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{9 d^{3/2} f^{3/2}}+\frac {2 b x \left (3 \left (\log \left (c x^n\right )-n \log (x)\right )-n\right )}{9 d f}+\frac {1}{9} b x^3 \left (3 \left (\log \left (c x^n\right )-n \log (x)\right )+3 n \log (x)-n\right ) \log \left (d f x^2+1\right )-\frac {2}{27} b x^3 \left (3 \left (\log \left (c x^n\right )-n \log (x)\right )-n\right )-\frac {2}{3} b d f n \left (-\frac {i \left (\text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+\log (x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )\right )}{2 d^{5/2} f^{5/2}}+\frac {i \left (\text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )+\log (x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )\right )}{2 d^{5/2} f^{5/2}}-\frac {x (\log (x)-1)}{d^2 f^2}+\frac {\frac {1}{3} x^3 \log (x)-\frac {x^3}{9}}{d f}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b x^{2} \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right ) + a x^{2} \log \left (d f x^{2} + 1\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )} x^{2} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.54, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right ) x^{2} \ln \left (\left (f \,x^{2}+\frac {1}{d}\right ) d \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{9} \, {\left (3 \, b x^{3} \log \left (x^{n}\right ) - {\left (b {\left (n - 3 \, \log \relax (c)\right )} - 3 \, a\right )} x^{3}\right )} \log \left (d f x^{2} + 1\right ) - \int \frac {2 \, {\left (3 \, b d f x^{4} \log \left (x^{n}\right ) + {\left (3 \, a d f - {\left (d f n - 3 \, d f \log \relax (c)\right )} b\right )} x^{4}\right )}}{9 \, {\left (d f x^{2} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,\ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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